Good observation, he is a king via acclamation, not force or bloodline (even though the Stark bloodline helped).
I would imagine The North is all about agnatic primogeniture (eldest legitimate male heir inherits).. which screws Jon Snow so many ways..
1) he's a bastard
2) as we have learned, he's not Eddard Stark's bastard.
3) his only Stark blood is from the cognatic (female) line.. Eddard's sister Lyanna.
4) with no legitimate male heirs (one could argue that since Bran is paralyzed and unlikely to father a child he is not longer legitimate.. you still have Eddard's brother Benjen to consider. I know, I know.. he took the vows to the Nights Watch.. wist.. what.. you mean just like Jon Snow did? That's right. But only Bran and Meera reed know the fiery-mace-wielding Benjen is still "alive". (hmmm.. maybe he is Azor Ahai?)
And if Agnatic-Cognatic Primogeniture rules the day.. then it gets worse for Jon Snow because
5) in the books.. Catelyn Stark has been raised from the dead and is kinda directing the Brotherhood without Banners in the Riverlands.
6) ignoring 5 for the TV show version.. Sansa and Arya would inherit before Jon... until they give birth to a male heir.. then he would be the Prince, I think, and they'd have to find a regent (which could be his mother) until he reaches a certain age.
7) and even if all those proper Starks died.. you have the cousin orphan Lord Robert Arryn of the Vale.. the one who likes to see people fly out the Moon Door.. the one controlled by Littlefinger... he would inherit as a legitimate Stark from the female line.... oh.. wait.. not he wouldn't.. he's half Tully, not Stark. My bad. So he might have a claim on the Riverlands.
I think you can see why there were so many wars of succession way back in the day.
But wait.. all is not lost for Jon Snow... if Rhaegar Targaryen secretly married Lyanna Stark while she was "abducted".. if they eloped.. then Jon Snow is legitimate... but Rhaegar was married to Elia martell (and The Mountain.. the Hound's brother Gregor Clegane.. murdered Rhaegar's whole family in Dorne.. which is why the Martell guy with the spear got his head crushed trying to avenge that. And Rhaegar was struck down by Robert Baratheon.
Damn I wish GRRM would finish that next book already!